rsos.royalsocietypublishing.org Research Cite this article: Huang L, Ni X, Ditto WL, Spano M, Carney PR, Lai Y-C. 2017 Detecting and characterizing high-frequency oscillations in epilepsy: a case study of big data analysis. R. Soc. open sci. 4: 160741. http://dx.doi.org/10.1098/rsos.160741 Received: 25 October 2016 Accepted: 22 December 2016 Subject Category: Engineering Subject Areas: biomedical engineering/complexity/ chaos theory Keywords: nonlinear dynamics, electroencephalogram, epileptic seizures, empirical mode decomposition, big data analysis, high-frequency oscillations Author for correspondence: Ying-Cheng Lai e-mail: [email protected]Detecting and characterizing high-frequency oscillations in epilepsy: a case study of big data analysis Liang Huang 1 , Xuan Ni 2 , William L. Ditto 5 , Mark Spano 3 , Paul R. Carney 6 and Ying-Cheng Lai 2,4 1 School of Physical Science and Technology, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China 2 School of Electrical, Computer and Energy Engineering, 3 School of Biological and Health Systems Engineering, and 4 Department of Physics, Arizona State University, Tempe, AZ 85287, USA 5 College of Sciences, North Carolina State University, Raleigh, NC 27695, USA 6 Pediatric Neurology and Epilepsy, Department of Neurology, University of North Carolina, 170 Manning Drive, Chapel Hill, NC 27599-7025, USA Y-CL, 0000-0002-0723-733X We develop a framework to uncover and analyse dynamical anomalies from massive, nonlinear and non-stationary time series data. The framework consists of three steps: preprocessing of massive datasets to eliminate erroneous data segments, application of the empirical mode decomposition and Hilbert transform paradigm to obtain the fundamental components embedded in the time series at distinct time scales, and statistical/scaling analysis of the components. As a case study, we apply our framework to detecting and characterizing high-frequency oscillations (HFOs) from a big database of rat electroencephalogram recordings. We find a striking phenomenon: HFOs exhibit on–off intermittency that can be quantified by algebraic scaling laws. Our framework can be generalized to big data-related problems in other fields such as large-scale sensor data and seismic data analysis. 1. Introduction Big data analysis [1–6], a frontier field in science and engineering, has broad applications ranging from biomedicine and smart health [7,8] to social behaviour quantification and energy optimization in civil infrastructures. For example, in biomedicine, 2017 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. on January 19, 2017 http://rsos.royalsocietypublishing.org/ Downloaded from
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ResearchCite this article: Huang L, Ni X, Ditto WL,Spano M, Carney PR, Lai Y-C. 2017 Detectingand characterizing high-frequency oscillationsin epilepsy: a case study of big data analysis.R. Soc. open sci. 4: 160741.http://dx.doi.org/10.1098/rsos.160741
Received: 25 October 2016Accepted: 22 December 2016
Subject Category:Engineering
Subject Areas:biomedical engineering/complexity/chaos theory
Keywords:nonlinear dynamics, electroencephalogram,epileptic seizures, empirical modedecomposition, big data analysis,high-frequency oscillations
Detecting andcharacterizinghigh-frequency oscillationsin epilepsy: a case study ofbig data analysisLiang Huang1, Xuan Ni2, William L. Ditto5, Mark
Spano3, Paul R. Carney6 and Ying-Cheng Lai2,4
1School of Physical Science and Technology, Lanzhou University, Lanzhou,Gansu 730000, People’s Republic of China2School of Electrical, Computer and Energy Engineering, 3School of Biological andHealth Systems Engineering, and 4Department of Physics, Arizona State University,Tempe, AZ 85287, USA5College of Sciences, North Carolina State University, Raleigh, NC 27695, USA6Pediatric Neurology and Epilepsy, Department of Neurology, University of NorthCarolina, 170 Manning Drive, Chapel Hill, NC 27599-7025, USA
Y-CL, 0000-0002-0723-733X
We develop a framework to uncover and analyse dynamicalanomalies from massive, nonlinear and non-stationarytime series data. The framework consists of three steps:preprocessing of massive datasets to eliminate erroneous datasegments, application of the empirical mode decompositionand Hilbert transform paradigm to obtain the fundamentalcomponents embedded in the time series at distinct timescales, and statistical/scaling analysis of the components.As a case study, we apply our framework to detecting andcharacterizing high-frequency oscillations (HFOs) from a bigdatabase of rat electroencephalogram recordings. We find astriking phenomenon: HFOs exhibit on–off intermittency thatcan be quantified by algebraic scaling laws. Our frameworkcan be generalized to big data-related problems in other fieldssuch as large-scale sensor data and seismic data analysis.
1. IntroductionBig data analysis [1–6], a frontier field in science and engineering,has broad applications ranging from biomedicine and smarthealth [7,8] to social behaviour quantification and energyoptimization in civil infrastructures. For example, in biomedicine,
2017 The Authors. Published by the Royal Society under the terms of the Creative CommonsAttribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricteduse, provided the original author and source are credited.
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................................................vast electroencephalogram (EEG) or electrocorticogram (ECoG) data are available for the analysis,detection and possibly prediction of epileptic seizures (e.g. [9–16]). In a modern infrastructure viewedas a complex dynamical system, large-scale sensor networks can be deployed to measure a number ofphysical signals to monitor the behaviours of the system in continuous time [17–19]. In a modern city,smart cameras are placed in every main street to monitor the traffic flow at all times. In a community, datacollected from a large number of users carrying various mobile and networked devices can be used forcommunity activity prediction [20]. In wireless communication, big datasets are ubiquitous [21,22]. In allthese cases, monitoring, sensing or measurements typically result in big datasets, and it is of considerableinterest to detect behaviours that deviate from the norm or the expected.
In this paper, we develop a general and systematic framework to detect hidden and anomalousdynamical events, or simply anomalies, from big datasets. The mathematical foundation of ourframework is Hilbert transform and instantaneous frequency analysis. The reason for this choice lies inthe fact that complex dynamical systems are typically nonlinear and non-stationary. For such systems, thetraditional Fourier analysis is limited because, fundamentally, they are designed for linear and stationarysystems. Windowed Fourier analysis may be feasible to generate patterns in the two-dimensionalfrequency–time plane pertinent to characteristic events, but two-dimensional feature identification isdifficult. By contrast, the features generated by the empirical mode decomposition (EMD) methodologyare one dimensional, which are easier to be identified computationally. The Hilbert transform andinstantaneous frequency-based analysis have proved to be especially suited for data from complex,nonlinear and non-stationary dynamical systems [23–25]. The challenge is to develop a mathematicallyjustified and computationally reasonable framework to uncover and characterize ‘unusual’ dynamicalindicators that may potentially be precursors to a large-scale, catastrophic dynamical event of the system.
The general principle underlying the development of our big data-based detection framework is asfollows. First, we develop an efficient procedure for preprocessing big datasets to exclude erroneousdata segments and statistical outliers. Next, we exploit a method based on a separation of time scales,the EMD method [23,24], to detect anomalous dynamical features of the system. Owing to its built-inability to obtain from a complex, seemingly random time series a number of dominant components withdistinct time scales, the method is anticipated to be especially effective for anomaly detection. We payparticular attention to the challenges associated with big datasets. Finally, we perform statistical analysisto identify and characterize the anomalies and articulate their implications.
As a concrete example to illustrate the general principle of our big data analysis framework, weaddress the detection of high-frequency oscillations (HFOs), which are local oscillatory field potentialsof frequencies greater than 100 Hz and usually have a duration less than 1 s [26–37]. Oscillations between100 and 200 Hz are called ripples and occur most frequently during episodes of awake immobility andslow-wave sleep. The HFOs in this range are believed to play an important role in information processingand consolidation of memory [38,39]. Pathologic HFOs (with frequency larger than 200 Hz, or fastripples [40]) reflect fields of hyper-synchronized action potentials within small discrete neuronal clustersresponsible for seizure generation. They can be recorded in association with interictal spikes only in areascapable of generating recurrent spontaneous seizures [41]. Thus detecting fast ripples can be useful inlocating the seizure onset zone in the epileptic network [29,42,43], and this was verified previously usingdatasets from a wide variety of patients [32]. In particular, it was found that almost all fast-ripple HFOswere recorded in seizure-generating structures of patients suffering from medial or polar temporal-lobeepilepsy, indicating that the ripples are a specific, intrinsic property of seizure-generating networks inthese brain areas. The pathologic HFOs and their spatial extent can potentially be used as biomarkers ofthe seizure onset zone, facilitating decisions as to whether surgical treatment would be necessary. Besidestheir role in locating the seizure onset zone, HFOs may also reflect the primary neuronal disturbancesresponsible for epilepsy and provide insights into the fundamental mechanisms of epileptogenesis andepileptogenicity [44,45].
Traditional methods such as the Fourier transform and spectral analysis assume stationarity and/orapproximate the physical phenomena with linear models. These approximations may lead to spuriouscomponents in their time–frequency distribution diagrams if the underlying signal is non-stationaryand nonlinear. EMD is a technique [23] to specifically deal with non-stationary and nonlinear signals.Given such a signal, EMD decomposes it into a small number of modes, the intrinsic mode functions(IMFs), each having a distinct time or frequency scale and preserving the amplitude of the oscillationsin the frequency range. The decomposed modes are orthogonal to each other, and the sum of all modesgives the original data. The ease and accuracy with which the EMD method processes non-stationaryand nonlinear signals have led to its widespread use in various applications such as seismic dataanalysis [23], chaotic time series analysis [24,46], neural signal processing in biomedical sciences [47],
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................................................meteorological data analysis [48] and image analysis [49]. We develop an EMD-based method to detectHFOs. Owing to its built-in ability to pick out from a complex, seemingly random time series a numberof dominant components of distinct time scales, the method is especially effective for the detection ofHFOs. We finally perform a statistical analysis and find a striking phenomenon: HFOs occur in an on–offintermittent manner with algebraic scaling. In addition to HFOs, our framework can detect populationspikes, oscillations in the frequency range from 10 to 50 Hz, as well as distinct and independent IMFs.
As pathologic HFOs reveal dynamical coherence within small discrete neuronal clusters responsiblefor seizure generation, a good understanding and accurate detection of HFOs may bring the grand goal ofearly seizure prediction one step closer to reality and would also improve the localization of the seizureonset zone to facilitate decision-making with regard to surgical treatment. Not only does our methodillustrate, in a detailed and concrete way, an effective way to analyse big datasets, our finding also haspotential impact in biomedicine and human health.
There were existing works on applying the EMD/Hilbert transform method to neural systems.Earlier the method was applied to analysing biological signals and performing curve fitting [50], and acombination of EMD, Hilbert transform and smoothed nonlinear energy operator was proposed to detectspikes hidden in human EEG data [51]. Subsequently, it was demonstrated [52] that the methodologycan be used to analyse neuronal oscillations in the hippocampus of epileptic rats in vivo with the resultthat the oscillations are characteristically different during the pre-ictal, seizure onset and ictal periodsof the epileptic EEG in different frequency bands. In another work [53], the EMD/Hilbert transformmethod was applied to detecting synchrony episodes in both time and frequency domains. The methodwas demonstrated to be useful for decomposing neuronal population oscillations to gain insights intoepileptic seizures [54], and EMD was used for extracting single-trial cortical beta oscillatory activitiesin EEG signals [55]. The outputs of EMD, i.e. the IMFs, were demonstrated to be useful for EEG signalclassification [56]. Our work differs from these previous works in that we address the issue of detectingHFOs and uncovering the underlying scaling law.
2. Results2.1. Pretreatment of datasetsHigh-sampling (12 KHz), multichannel (32–64 channels), continuous recordings of local field potentialsin freely moving rodents present unique technical challenges. Although most channels continue to recordover a four- to six-week period, over time the integrity of the signal degrades and electrode recordingmay come off- and online. To this end, it is important to preprocess data files to exclude gaps in data. Thisin itself is challenging due to the large size of each dataset (approx. 5 TB), variability during recordingsof local field potentials, and gaps in data. Here, we develop a fully automated statistical method. Theresulting ‘data-mining’ algorithm is general and we expect it to be useful for dealing with other massivedatasets.
For our study, we examine EEG data taken from a rat model of the approach to epilepsy. The typicalsize of a binary file in our database is about 600–700 MB. Each file belongs to a certain channel (specifiedby a channel number) and a specific time duration (specified by a file number). We regard the channel andfile numbers as two orthogonal dimensions and plot the contour of a suitable statistical quantity (to bediscussed below) in the two-dimensional plane, so the data of one rat (approx. 5 TB) can be representedby a single contour plot. The whole process can be programmed to be highly parallelized, providing aglobal overview of the raw EEG data.
Let di, (i = 1, . . . , L) be the value of the EEG signal for a single sample, where L is the number ofsamples in a binary file. In the experiment, each value di is recorded as a 16-bit integer, so di ∈ [−N, N − 1],where N = 215 and, typically we have L ∼ 3 × 108 samples. We then examine the values of di and countthe number of each value present in the file, which results in an array nj, j = −N, . . . , N. Repeated valuesover some periods in the oscillation pattern lead to corrupted files, which can be due to recording errors—this, indeed, happened in our experiment. In general, when hours or even minutes of bad recordings ofzeros are encountered, the number n0 of zeros in the file counted will increase rapidly. These features ofabnormal recordings will be utilized to exclude the corrupted files.
Figure 1 shows four typical types of nj distributions obtained from channel 02 of Rat004 composedof 229 data files. For binary files that have large numbers of continuous zeros, for example, file number20, the distribution is shown in figure 1a. An example of a corrupted file is where a specific pattern ofoscillations is embedded repeatedly in most of the data in the file. The corresponding distribution is
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Figure 1. Pretreatment of massive data from rat EEG. Different types of distributions of nj for Rat004 channel 02. For each panel, the y-axis is normalized by themaximumof nj . The four panels correspond to: (a) a corrupted filewith a large number of zeros (file no. 20), (b) abad recordingwith repetitions of oscillating patterns (file no. 73), (c) a normal file without transitions (file no. 77) and (d) a file containinga seizure (file no. 99).
shown in figure 1b. The distributions from normal data files qualified for dynamical analysis are shownin figure 1c,d. The distribution in figure 1c is approximately Gaussian. However, seizure events can causedistortions from the Gaussian distribution, as evidenced by figure 1d for file number 99 where a clinicallycertified seizure is present. We observe that the distribution becomes somewhat narrowed (as comparedwith the case of no seizure) and slightly asymmetrical.
After obtaining the distribution for each file, we define and compute a statistical quantity for eachfile, and assemble the files within the same channel according to this quantity, as follows. Let sk = nk−N −nk−N−1 (k = 1, . . . , 2N), where sk represents the difference between two neighbouring counts, and let σ 2
s =1/(2N − 1)
∑2Nk=1(sk − s̄)2 be the variance, where s̄ is the mean value of sk. Note that nj is not normalized
and their sum is the data length L of the file. Denoting 〈n〉j as the smoothed curve of nj, we have 〈n〉j ∼ L.The fluctuations, as characterized by sk, are in general proportional to 〈n〉j. As a result, the variance σ 2
s ispositively correlated with L, e.g. a larger value of L would result in a larger value of σ 2
s .Thus for small files that are normal in all other aspects, we will have smaller σs values, which can be
clearly identified from figure 2 as the points below the majority. This figure also shows some points withextremely large σs values. These points correspond to the binary files with large numbers of continuouszeros (figure 1a). The corrupted data all have σs values in the range 104 ∼ 105 (figure 1b) which canbe excluded readily, as shown in figure 2. A transition in σs at file number 99 (when the first seizureoccurred) is observed.
By applying the same procedure to multiple channels, the massive EEG data from one rat can beexpressed using a single contour plot of σs, as shown in figure 3. We can immediately identify differenttypes of data in terms of the values of log10 σs. It should be noted that for different ‘abnormal’ situations,e.g. small files, contamination with zeros or corrupted data, there can be different methods of remedybased on examining different aspects particular to the data. However, our method is general and efficientin that a single indicator is effective at distinguishing the different types of abnormalities in the filesduring the preprocessing stage of the massive database for further dynamical and statistical analysis.
2.2. Empirical mode decomposition analysis of electroencephalogram dataWe conducted extensive tests of applying the EMD procedure to EEG data (see Material and methods). Ageneral finding is that the resulting IMFs in different frequency ranges possess statistical features that are
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Figure 2. Statistical properties ofmassive data from rat EEG. Standard deviationσs for Rat004 channel 02. Red circles denote the normalfiles; green squares are the files with large numbers of zeros; blue crosses are corrupted files; pink diamonds are small files; cyan trianglesare small files withmany zeros; black star is small corrupted files. The arrowmarks the file 99 which has the first seizure. The inset showsthe enlarged area around file 99 on a linear scale.
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Figure 3. Contour representation of massive data from rat EEG. Contour plot of log10 σs for Rat001. (a) The whole σs range. Differenttypes of data are classified according to the value of log10 σs. In (b), the contour is for values of 102 ≤ σs ≤ 103 (good data), and theremaining values ofσs are set to 50 so that the dark blue area marks all abnormal data.
relevant to certain brain activities, demonstrating that the EMD methodology can be effective for probingthe dynamical origins of epileptogenesis. For example, typically the frequencies of the first 5 IMFs areabout 5 kHz, 2 kHz, 1 kHz, 500 Hz, 200 Hz, 100Hz. As the sampling frequency is 12 kHz, the first threemodes correspond to mostly noise contained in the original EEG data. The fourth to sixth modes, whosefrequencies lie in the range between 50 Hz and 800 Hz represent the intrinsic dynamical evolution of theunderlying brain system.
Our procedure for analysing long EEG data thus consists of performing the EMDs to obtain differentIMFs, calculating the amplitudes and frequencies of the IMFs that are deemed to reveal the dynamicalevolution of the brain, and performing statistical analysis of the on-intervals for the IMFs in a properfrequency range. An example is shown in figure 4, where the distributions of the amplitude andfrequency of an IMF for a particular channel of two months’ duration are shown. The entire dataset wasdivided into 230 files, each containing 7 h of EEG recording. Note that when performing the EMD, thedata are broken into small segments, e.g. 5 s for each segment, to make the computation more efficient.To calculate the IMFs, a 0.5 s segment is included at each end of the 5 s segment to eliminate the edgeeffect so that the IMFs can be accurately determined. From figure 4, we can distinguish the changes inthe rat brain activity, such as stimulation and the occurrence of the first seizure. Recurrent seizures arenot so clearly visible in this plot. Another apparent feature revealed is the circadian periodicity. The EEGrecording has a 24 h periodicity because of the circadian activity or of the external treatment of the rat
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Figure 4. Typical EMD representation of massive rate EEG data. (a) Contour plot of normalized distribution of amplitude A (in arbitraryunits) varying in time of a particular EMD mode of interest (IMF5, approx. 200 Hz) for channel 11 in CA1 of EEG recording of a rat overa two-month period. The all-blue region indicates corrupted files. Each file is a 7 h recording at the sampling frequency 12 kHz. Thus,the vertical axis ‘file no.’ indicates time. The distribution is calculated and then normalized by the maximum value for each file. The ratunderwent surgery between file 28 and file 29, and the first seizure occurred in file 99, as indicated by the red arrows. The comb-likestructure indicates the circadian periodicity. (b) Normalized distribution of the frequency f of the mode.
such as feeding, etc., which also changes the frequency and amplitude of each decomposed IMF. As eachfile is 7 h long, the circadian periodicity indicates a periodicity of three (or four) files in the plot, which isapparent from the comb-like structure in the plot, especially in figure 4a, where two adjacent comb teethhave a separation of about three files.
2.3. Detection of high-frequency oscillations and population spikesTo illustrate the procedure of detection of HFOs and population spikes, we reduce the sampling rate sothat these dynamical events can be visualized clearly. Note that, when the sampling rate is reduced, thenoisy components are effectively filtered out, so the first few IMFs become important. (In the detectionand statistics of HFOs and population spikes in the following sections, higher sampling frequency shouldstill be used, in which case the first few IMFs need to be disregarded, as discussed above). Figure 5 shows,for a segment of down-sampled EEG data, the relevant empirical modes. For this dataset, the frequenciesare 200–500 Hz for mode 1, 80–200 Hz for mode 2, about 50 Hz for mode 3 and 30 Hz for mode 4. Takingmode 2 as an example, the IMF will have small amplitude if the original EEG data does not containoscillations in the corresponding frequency range. When the EEG data contains these oscillations, theywill be revealed in the corresponding IMF. As HFOs are generally associated with frequencies largerthan 80 Hz, they will be revealed in the first two modes. The population spikes (with a time scale of0.1 s) are decomposed by EMD into oscillations in the frequency range 10–50 Hz, thus they will mainlybe manifested in modes 3 and 4. The EEG data in figure 5a contains an HFO and a population spike.It is apparent that the HFO and population spike are separated by EMD into different modes and arelocalized in different time scales, e.g. figure 5c for the HFO and figure 5d,e for the population spike.Thus, the amplitude of the modes evolving in time can be used to detect the HFOs or population spikes,depending on the frequency range of the mode.
Our results thus suggest strongly the feasibility of developing EMD-based algorithms tosystematically detect all the HFOs and population spikes. In this regard, we note an existing methodof detection of HFOs, which employs short-time energy or line length of the acquired data for HFOsin some small frequency ranges [57]. Our method is capable of detecting HFOs and can be used todistinguish various oscillation profiles. Based on the detected HFOs and population spikes, extensivestatistical analyses for the five critical phases during epileptogenesis, namely, pre-stimulation state, pre-seizure state, status epilepticus phase, epilepsy latent period, spontaneous/recurrent seizure period, canbe carried out to gain unprecedented insights into epileptogenesis.
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Figure 5. Example of EMD-based HFO detection from EEG data. (a) A 1.5 s segment of normalized EEG data containing an HFO anda population spike. (b–e) The IMFs in the frequency range of interest. The HFO is revealed in IMF 2 and the population spike is revealedin IMF 3 and IMF 4.
It can occur that, for a particular signal whose highest frequency component is most significant to theunderlying dynamics, the first IMF contains the dominant dynamics with the highest frequency, the nextis a lower frequency background to it and so on. However, while the first IMF is the highest frequencycomponent, the corresponding frequency range may not necessarily be relevant to the system dynamics.In fact, we found that typically the first IMF corresponds to noise, and the next IMF contains informationabout the dynamics of the system. Which IMFs are actually useful and informative depends on thenature of the original signal. More specifically, what EMD does is to decompose the signal into differentfrequency components through different IMFs, which contain the time varying amplitude and frequencyinformation for each component embedded in the original signal. If the signal is contaminated bynoise, the first IMF would be the noise component that contains little information about the underlyingdynamics. Our analysis of the massive EEG data indicates that this is indeed the case.
2.4. Automated detection and classification of high-frequency oscillationsOur method to detect, characterize and understand HFOs from EEG recordings consists of the followingsteps: (i) performing EMD and calculating distinct IMFs, (ii) searching for HFOs based on the amplitudesof IMFs and (iii) classifying HFOs in terms of their frequencies and calculating the statistical propertiesof HFOs. An illustration of the steps is shown in figure 6.
We first calculate the amplitudes of the IMFs from an automated EMD procedure. We then locatethe extrema of one IMF and define the interval between two neighbouring maxima (or minima) to beone period T, as shown in figure 6a. Unlike the Fourier transform that becomes ineffective in time-seriesanalysis when the signal frequency changes with time, EMD is well suited for generating IMFs whosefrequencies vary with time, i.e. when the period is a function of time: T = T(t). We set a moving timewindow of size w and calculate the average IMF amplitude within the window. The window containsa fixed number of IMF periods. As the period varies with time, the actual time span of the window
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Figure 6. Illustration of HFO detectionmethod. Themethod consists of three steps. (a) Computing the amplitude function fromeach IMFgenerated by EMD. The size of the moving window isw = 7 periods (indicated by the blue dashed boxes). The time step for the movingwindow is�w. (b) For each IMF, we locate the on-intervals, find HFOs, and combine adjacent HFOs if they are too close to each other.The blue dashed line is the threshold chosen for the segment of the amplitude function. (c) Classifying HFOs in terms of their frequencies,e.g. ripples (solid blue triangles), fast ripples (open magenta triangles) and then combining overlapping HFOs across different IMFs, asshown in the blue dashed box.
also changes with time. As an example, we show in figure 6a two windows wm and wn, each containingthe same number (7) of IMF periods. Apparently, their sizes are different. The amplitude values areweighted magnitudes and are calculated as Am = (1/2)
∑i∈wm
(xi+1 − xi)(|yi+1| + |yi|), where xi and yiare the position and magnitude of the IMF, respectively, and the factor (xi+1 − xi) is the correspondingweight. The calculation is repeated when the window is moved to the next position by the step size �w,which is also chosen to contain a certain number of IMF periods. Small values of w and �w result inrapidly oscillating amplitude functions, whereas too large values of w and �w would cause a loss ofinformation. Empirically, the window size can be chosen to include several IMF periods, e.g. six to nine,where the moving forward step size �w can be set as one.
The next step is to find on-intervals, time intervals when the amplitude values are larger than a certainthreshold Ac, as shown in figure 6b. To set a proper threshold, we can pick a segment (e.g. 1 h long)from the amplitude data and calculate the mean μ as well as the standard deviation σ . One way to setthe threshold is Ac = aμμ + aσ σ , where aμ and aσ are two adjustable parameters on the order unity. Tocharacterize each on-interval Oi, we define an on-area Si, the area of the IMF in the on-interval abovethe threshold:
Si = (1/2)∑j∈Oi
(xj+1 − xj)(Aj+1 + Aj − 2Ac), (2.1)
where xj and Aj are the position and magnitude of the amplitude function of the underlying IMF,respectively. The notation Si with the capital S should be distinguished from sk that denotes the differencebetween two neighbouring counts. On-intervals are sorted in the descending order in terms of their on-areas: S1 > S2 > · · · . The HFOs are identified as those with the largest on-areas, i.e. with longer durationand larger oscillating amplitudes. As the on-area values of HFOs are typically much larger than thoseof non-HFO on-intervals, the HFOs can be reliably identified as the outliers in the on-area statistics,as follows.
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Figure 7. Example of successful HFO and PS detection. (a) Original EEG data plot of about 3 s. (b) IMF 5 plot with solid blue trianglesmarking the ripples and open magenta triangles marking the fast ripples. (c) The amplitude of IMF 5. The horizontal blue line isthe threshold for separating on/off intervals of HFOs. The threshold is calculated from the amplitude data segment of about 1 h. Thecomputational parameters are aμ = 1 and aσ = 1. (a′)–(c′) The original data, IMF 6, and its amplitude function, respectively. The blackdiamonds mark the position of the population spikes.
Starting from the most significant on-interval O1, we can evaluate its on-area S1 with a score functionof the remaining on-intervals. If
S1 > αE[{Si}i≥2] + β
√Var[{Si}i≥2], (2.2)
O1 will be identified as an HFO, where α and β are two adjustable parameters, and E[·] and Var[·] arethe expectation and variance functions. This process is repeated until no on-intervals in the remainingsequence satisfy the criterion. Typically, for one IMF, approximately 10% of the on-intervals would beselected as HFOs. Among the remaining HFOs, one can see that some are so close to each other thatit is reasonable to combine them. The combinations are carried out wherever the gap G between twoneighbouring HFOs is small, i.e. when G < g · min(THFO1 , THFO2 ), where g is the parametric gap toleranceratio and THFO is the HFO duration. An example of combining HFOs is shown in figure 6b.
The HFOs in different frequency ranges are usually responsible for different brain behaviours such asnormal information processing and spontaneous seizures. The third step of our procedure is to calculatethe frequencies of various HFOs, which is done by locating the starting and ending times of an HFO,finding the number n of oscillating periods within it and dividing by its duration: f = n/THFO. Whennecessary, we combine overlapping HFOs across different IMFs, as shown in figure 6c. When variousHFOs have been identified, we can classify them into distinct frequency ranges: low-frequency range(less than 80 Hz), ripples (80–200 Hz, between pairs of solid blue triangles in figure 6c) and fast ripples(greater than 200 Hz, between pairs of open magenta triangles in figure 6c). HFOs of frequencies lowerthan 80 Hz are identified as population spikes. An example of identifying and classifying HFOs is shownin figure 7.
2.5. Statistical and scaling properties of high-frequency oscillationsAs described, the on- and off-intervals associated with HFOs can be determined by setting a thresholdvalue Ac in the amplitude function of a given IMF. For a given HFO, an on-area can then be defined,which is the area of the portion of the amplitude function above the threshold. The most significantHFOs are those with the largest on-areas. The on-intervals thus provide a base for characterizing theHFOs. Intuitively, HFOs of various magnitude correspond to ‘islands’ of various sizes above the ‘sea’level defined by the threshold. To obtain a more complete understanding of HFOs, it is insightful to
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Figure 8. Determination of threshold Ac. Normalized distribution P(A) of the amplitude A for files 1–28 for the samemode as in figure 4,where Ap is the value of amplitude at the peak of the distribution. For example, by setting P(A)= 0.1 and assuming that Ac > Ap, Ac canbe determined to be 61. If P(A) takes a smaller value, then Ac will be larger.
examine the corresponding ‘undersea’ dynamics (below the threshold). It is computationally feasible tostudy the dynamical and statistical characteristics of the oscillations below the ‘sea’ level but only withina certain depth.
To illustrate our approach in a concrete way, we take the example in figure 7 and focus on mode 5because the frequency of this mode lies in a suitable HFO frequency range. From the contour plot of thedistribution of amplitude versus file number, figure 4, we see that there are several regions of distinctproperties. In particular, the EEG data are relatively stable before the stimulation, say between files 28and 29. After the stimulation, the data changed characteristically, as can be seen from the amplitudedistribution plot in figure 4a (indicated by the arrow between files 28 and 29). The first seizure occurredin file 99, and the data are stable for files 29–99. There is a relatively small change between files 51 and52, as can be seen from the amplitude plot in figure 4a, when the rat was actually moved from one cageto another. We have checked other modes and also data from other channels and found a consistency inthe specific data segmentation as described. It is thus useful to study these different segments separately:files 1–28 (before stimulation), files 29–51 (between stimulation and first seizure), files 52–94 (pre-ictalphase), files 100–172 (postictal phase but with recurrent seizures in files 105 and 125) and files 175–223(with recurrent seizures in files 192, 209 and 216). To obtain stable statistics, we temporarily disregard afew files that are in the transitional regime. However, these files may be important in providing possiblehints about how the seizure (e.g. the one that occurred around file 99) is developed in terms of thetransient dynamics. The specific segmentation scheme of the EEG data is only for one rat, but it is validfor all the channels.
For different segments the signal amplitudes can have systematic differences, and in certain casesthe average amplitude can be, for example, twice larger in one segment than in another. It is thusnecessary to determine and set different thresholds in different segments. To do this, first we calculatethe normalized distribution of the amplitude A in each segment. The distribution for files 1–28 isshown in figure 8. Second, from the peak point defined as P(Ap) = 1 (as P is normalized to unity), wedecrease P with small steps, e.g. P = 0.98, 0.96, . . . , 0.02. For each P value, we determine the correspondingthreshold Ac (>Ap), as demonstrated in figure 8 for P = 0.1, where the corresponding threshold isAc = 61. Third, for each value of Ac, we calculate all the on-intervals in this segment from the amplitudefunction. The distributions of the durations T of the on-intervals from different segments are thencompared, and various threshold values are set such that the values of P(Ac) are all identical acrossthe segments.
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Figure 9. Statistical and scaling behaviours of HFOs. Distributions of on-interval T of IMF 5 (channel 11, figure 4): (a–e) for files 1–28,29–51, 52–94, 100–172, 175–223, respectively. The numbers of on-intervals are 344 310, 314 698, 431 674, 498 947 and 510 096 for (a–e),respectively. An algebraic distribution is observedwith different exponents for different segments. The exponent for the solid, dotted anddash-dotted lines are−3.7,−4.5 and−5.5, respectively. The threshold Ac is chosen such that P(Ac)= 0.02 for all the segments.
Figure 9 shows an example of the statistical distribution of the on-intervals for all five segmentsas specified above. We observe algebraic distributions. For each segment, the algebraic scaling regimeextends over at least one order of magnitude in the length of the on-interval. The distributions for thethree segments before the first seizure have approximately the same algebraic scaling exponent, as shownin figure 9a–c, although the amplitude value varies continuously for these segments, as shown in figure 4.After the first seizure, the exponent becomes smaller, indicating more on-intervals with longer durations.For example, in figure 9d, for on-intervals with T ∼ 0.4 s, the probability is 100 times larger than thosebefore the seizure. For files 175–223, however, the exponent is somewhat increased, signifying a decreasein the probability of longer on-intervals, but it is still larger than the probability before the seizure. Wehave systematically checked on-interval statistics for different thresholds. The algebraic scaling and thequalitative difference among the scaling exponents from different segments are robust with respect tovariations of the threshold in a certain range (e.g. P(Ac) ranging from 0.02 to 0.3).
As many on-intervals (especially the long ones) correspond actually to HFOs, our discovery of thealgebraic scaling suggests that the HFOs appear more active and sustaining associated with seizureactivities, which is consistent with previous observations [29,31,32]. In general, an algebraic scalingindicates a hierarchical organization in the underlying dynamics, which in our case, suggests such anorganization in the brain neuronal activities. For example, the local synchrony among discrete neuronclusters may vary in hierarchical scales. The fact that approximately the same algebraic scaling exponentoccurred before the seizure indicates that, after the stimulation, while evolving toward epilepsy, theunderlying dynamics behave the same as in the normal brain. This could be due to the latency effect ofthe stimulation. The development into epilepsy, however, occurs in a relatively short period, similar tothe cascading phenomena associated with earthquakes [58].
We have also checked other channels, which are so selected that they belong to different (neural)correlation clusters. For some channels, behaviours similar to those in figure 9 are observed, butsignificant deviations occur for some other channels. This may be because the HFO and the seizureonset zone are usually highly localized [32]. As a result, only within a proper range of this zone can theHFOs be detected. As the distance between the neighbouring channels is quite small (approx. 0.25 mm),the HFO and the underlying neuronal activity could be revealed in a small subset of channels.
We find that the algebraic scaling law for the on-intervals of HFOs holds for all the animal models.Figure 10a–f shows more examples for a different rat. In particular, the scaling was calculated for aspecific channel for the pre-stimulation state, post-stimulation state, evolution towards seizure, the statusepilepticus phase, the epilepsy latent period and the spontaneous/recurrent seizure period, for panels(a–f ), respectively. We see that, while the details of the scaling behaviours can be different for the distinctcritical phases (e.g. during epileptogenesis the on-intervals with longer durations are dominant after thefirst seizure), the algebraic nature of the scaling law is robust.
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Figure 10. Statistical and scaling behaviours of HFOs, more examples. Distributions of on-interval T of IMF 5 (channel 6 of rat 9): (a–f )for files 1–19, 32–57, 63–72, 78–97, 98–118 and 119–149, corresponding to the pre-stimulation state, post-stimulation state, evolvingtowards seizure, status epilepticus phase, epilepsy latent period and spontaneous/recurrent seizure period, respectively. The numbers ofon-intervals are 354 499, 561 669, 300 291, 458 649, 293 118 and 438 919 for (a–f ), respectively. An algebraic distribution is observedwithdifferent exponents for different segments, where the exponents are−5.7,−3.3,−2.9 for dash-dotted line, solid line and dotted line,respectively. The criterion for choosing the threshold Ac is the same as in figure 9.
3. DiscussionSeizure prediction, early recognition and blockage of seizures are considered by the membershipof the American Epilepsy Society (AES) as the first research priority listed among 15. To achievethese goals a good understanding of the origin, mechanism and dynamics of seizures is necessary.At present the only accessible avenue to probe the origin of epileptic seizures is multiple-channelEEG or ECoG data. Continuous improvement in the experimental methodology has made such datahighly reliable and generally of high quality. A challenge is that the amount of EEG or ECoG data ismassive. An issue of paramount importance and significant interest is to extract knowledge about epilepsyfrom data.
We have developed a method to detect, characterize and analyse the dynamical behaviour of HFOsfrom a massive database of extensive EEG recordings of a number epileptic rats over two months. Wefirst devise a general and efficient procedure for preprocessing the massive database to exclude erroneousdata files and statistical anomalies. (The procedure should also be applicable to massive datasets of othersorts, such as large-scale sensor data or seismic data collections.) We then articulate a procedure basedon separating the time scales, the EMD method [23,24], to detect HFOs. We finally perform a statisticalanalysis and find evidence for a striking phenomenon: the occurrences of HFOs appear in an on–offintermittent manner and the time intervals that they last exhibit an algebraic scaling.
The methods and results in this paper can potentially be extended to other fields. Big datasetsarise not only in biomedicine but also in other fields of science and engineering. For example, in civilengineering, large sensor arrays are often employed to monitor the temperature, humidity and energyflows in large, complex infrastructure systems in a continuous-time fashion. In such an application,the underlying system is in general non-stationary and nonlinear, and to detect behaviours thatdeviate from the norm or the expected is of considerable interest. Another example is earthquakedata. A previous work [58] indicated that seizures can be regarded as ‘quakes of the brain’. It is thenconceivable that the idea, method and algorithms developed in this paper can be extended to bigseismic database for detecting anomalous oscillating signals (similar to HFOs) preceding the actualoccurrence of earthquakes. In general, the EMD/Hilbert transform-based methodology demonstratedin this paper has broad applications because it is specifically designed [23] to deal with nonlinearand non-stationary systems. Owing to its built-in ability to obtain from a complex, seemingly randomtime series a number of dominant components of distinct time scales, the method is anticipatedto be especially effective for anomaly detection. A challenge is to develop mathematically justified,computationally reasonable and automated procedures to detect anomalies, from big datasets, which
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................................................has been addressed in this paper through the detection of HFOs from massive EEG data with multipleanimal models.
We now discuss a number of issues that may warrant future efforts.First, there is room to improve the EMD-based algorithms developed in this paper, leading possibly
to a fully automated method for detecting HFOs and population spikes from massive data for all distinctepileptic stages including pre-stimulation, pre-seizure and recurrent seizures. This will provide a basefor probing into the emergence and evolution of HFOs through detailed analyses using methods fromnonlinear dynamics, statistics and statistical physics. Special features associated with different typesof brain activities can be identified, with the grand goal to exploit the predictive power of HFOs forepileptic seizures. Many questions, which are previously unimaginable, can be asked. For example, doesa general class of HFOs exist, regardless of the specific brain activities? What types of HFOs are especiallyrelated to seizures? How localized are they around the regions of seizure onset? Are there systematicand characteristic changes in the HFOs during the ictal phase? What types of HFOs are associated withrecurrent seizures and is there any relation to the latency period? Answers to those questions and manymore will provide a comprehensive picture for the dynamical role of HFOs in epileptogenesis.
Second, with our optimized EMD-based method, HFOs and population spikes can be detectedreliably for all the channels. The issue of spatiotemporal evolution of these dynamical events in thebrain can then be addressed. For example, we have observed that certain population spikes appearonly in some channels at a given time, i.e. they may be highly localized in space. Some HFOs can,however, occur in many channels simultaneously. A possible reason for the dispersive character ofthe HFOs is that the distance between neighbouring recording sites is about 0.25 mm, which can bewithin the propagating range of the underlying neuronal activities that cause the pathologic HFOs. Theavailable database from a dense electrode grid thus provides a useful probe to study the propagation ofneuronal synchronous activities. For example, by examining the exact timings of HFOs and populationspikes and the occurrence of the epileptic seizure in different EEG channels, the propagating patternof these dynamical events can be determined and, consequently, the sources triggering these eventsmay be identified. A mapping between the epileptic dynamics and activities in different brain regionscan be made and the temporal evolution of the mapping can be studied. It would also be useful toexamine the correlation patterns of the distinct dynamical events and compare them with those of thebackground neuronal activities. All these have the potential to provide deeper insights into the origin ofepileptogenesis.
Third, in this paper we focused on stable states of the brain, which are the states that last for relativelylong periods of time. The transient behaviours were neglected. It would be interesting to study thenonlinear and complex transient dynamics [59] associated with epileptogenesis and brain behavioursin general.
Fourth, there were previous studies of exploiting nonlinear dynamics for analysing epilepticseizures (e.g. [60–66]), and on–off intermittency is an ubiquitous phenomenon in nonlinear dynamicalsystems [67–82]. It is known in nonlinear dynamics that on–off intermittency can be controlled byapplying small but deliberately chosen perturbations to the system [83]. The power-law statistics ofthe on-intervals associated with HFOs indicate the emergence of a hierarchical organization in theneuronal activities. However, it is difficult to determine uniquely the underlying dynamical mechanismthat is distinct from the well-documented mechanism for on–off intermittency. Nonetheless, there arecommon features between the intermittent behaviours of HFOs uncovered in this paper and thoseof on–off intermittency, such as scaling properties. If a specific type of HFO can be correlated withthe occurrence of seizures and if the underlying dynamics bear similarity to those of generic on–offintermittency, it may be possible to investigate controlling HFO dynamics based on previous works oncontrolling on–off intermittency. In particular, can the on–off intermittent dynamics of the epileptic HFOsbe controlled by using, say, small and infrequent brain stimuli to delay or even eliminate seizures? Theresults in this paper provide a base for developing computational and experimental schemes to test thiscontrol idea.
4. Material and methods4.1. Setting of experimental data collectionExperiments were performed on five, two-month old, male Sprague Dawley rats, each weighing between210 and 265 g. The rats were induced into the state of complete anaesthesia by subcutaneous injection of
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................................................10 mg kg−1 (0.1 ml by volume) of xylazine and maintained in the anaesthetized state using isoflorane (1.5)administered through inhalation by a precision vaporizer. For each rat, the top of its head was shavedand chemically sterilized with iodine and alcohol. The skull was exposed by a mid-sagittal incision. Inthree of the five rats, a bipolar, twisted, Teflon-coated, stainless steel electrode (330 µm) was implanted inthe right posterior ventral hippocampus (5.3 mm caudal to bregma, 4.9 mm right of midline suture andat a depth of 5 mm from the dura) for stimulating the rat into status epilepticus. The remaining two ratsare for control. In all the five rats, a 16 microwire (50 µm, TDT Technologies, Alachua, FL, USA) electrodearraying two rows separated by 500 µm with electrode spacing of 250 µm, was implanted to the left ofmidline suture horizontally in the CA1–CA2 and dentate gyrus of the hippocampus. The furthest leftmicrowire was 4.4 mm caudal to bregma, 4.6 mm left of midline suture and at a depth of 3.1 mm fromthe dura. A second microwire array of 16 electrodes was implanted to the right of midline suture ina diagonal fashion. The furthest right microwire was 3.2 mm caudal to bregma, 2.2 mm to the right ofmidline suture. The closest right microwire was 5.2 mm caudal to bregma, 1.7 mm to the right of midlinesuture and at a depth of 3.1 mm from the dura. Finally, four 0.8 mm stainless steel screws were placed inthe skull to anchor the microwire electrode array: two screws were AP 2 mm to the bregma and bilaterally2 mm and served as the ground electrodes while two screws were AP −2 mm to the lambdoidal sutureand bilaterally 2 mm and served as the reference electrodes. The entire surgical area was then closed andsecured with cranioplast cement. Following surgery the rats were allowed to recover for a week.
4.2. Data acquisition and file structuresElectrophysiological recordings were conducted by hooking each rat onto a 32-channel commutator, theoutput of which was fed into the recording system comprising two 16-channel pre-amps, which digitizedthe incoming signal with a 16 bit A/D converter at a sampling rate of 12 kHz (approx. 12 207 Hz). Thedigitized signal was then sent over a fibre optic cable to a Pentusa RX-5 data acquisition board (TuckerDavis Technologies). The digital stream of data was stored for later processing. For each channel, thedata were recorded and saved as 16-bit signed integer binary files, each of size 600–700 MB (approx. 7.5 hrecording time). Thus, for each rat, there were 32 channels, each of which has between 153 and 317 filesdepending on the recording durations. Each binary file was assigned a rat number, a channel number anda file number. The data were recorded over two months for most of the rats, including pre-stimulationstage, pre-seizure stage, status epilepticus phase, epilepsy latent period and spontaneous/recurrentseizure period.
The sampling rate of the data was relatively high, making it possible to analyse high-frequency, short-duration dynamical events in the brain such as HFOs and population spikes. The typical duration ofan HFO is about 100 ms and its characteristic spontaneous frequency can be as high as several hundredhertz. In such a case, our data will have 40 points for a single oscillation period, which is generallyenough for various analyses of HFOs. The extensive database provides us with a platform to comparethe high-frequency events in different stages in the evolution of epileptic seizures and to systematicallyinvestigate the dynamics of epileptogenesis.
4.3. Empirical mode decomposition of electroencephalogram dataEMD is specifically designed to deal with nonlinear and non-stationary datasets. In particular, EMDdecomposes the signal into a series of IMFs, as follows. For a given signal, EMD determines the localmaxima and local minima, and connects them with cubic splines to form an IMF. One then subtracts theIMF from the original signal, and repeats the process to get the second IMF and so on. The procedureis repeated until the remaining signal becomes monotonic. The IMFs are orthogonal to each other (atleast locally) and their sum restores the original data. Thus, effectively, the original signal has beendecomposed into the IMFs, each in a distinct frequency range, whose non-stationary amplitude andfrequency information is well preserved.
Ideally, for given EEG data, the EMD method returns a set of IMFs in separate frequency ranges.Practically, as each data file is too large to be processed computationally, we need to divide the data intosmall segments so that each segment can be computed within the memory of our computers. To deal withthe boundary effect properly, for each data segment, we include an extra but much smaller subset of datapoints on both ends of the segment, which are from neighbouring segments. These are the correspondingboundary sets. After performing the EMD calculations, only the IMFs within the original data segmentare kept, while those associated with the boundary sets are discarded. For a given data segment, theresulting IMFs usually depend on the choices of the sizes of the segment and the boundary sets. In
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Figure 11. A demonstration of adding small oscillations in EMD computation to eliminate spurious large values in IMFs. (a) A 5 s segmentof data with about 0.4 s zeros, as indicated by the dotted circle. (b–f ) The first 5 IMFs directly calculated from the data in (a). (g) Data (a)with added small oscillations on the scale of unity (see text), which is almost invisible from the figure. (h–l) The first 5 IMFs calculatedfrom the data in (g). Insets of (g) and (k) show magnification of the zero region. Note that the scale of the y axis is much larger in (b–f )than those in (h–l). The anomalies appeared in (b–f ) are effectively removed by the simple method of adding small oscillations to thedata segment.
particular, the larger the boundary sets, the more accurate the IMFs, but the amount of the computationwill also increase. A systematic test on varying sizes of the boundary sets indicates that the choice of 0.5 sduration (corresponding to 6103 data points at the recording sampling frequency) for each boundaryset yields accurate IMFs with tolerable extra computation load. The limited computational power alsostipulates that the size of each segment itself cannot be too large. Our systematic test gave 5 s (approx.61 035 data points) as the optimal duration for balancing the computation time and the reliability of theresults. We use the code developed by Rilling [84] to perform the EMD calculations by modifying theoriginal C-Matlab interface to C-codes.
Another practical issue is that the data may contain some discontinuities. In such a case, theEMD program may diverge or have abnormally large values (figure 11a–f ). A remedy is to add
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................................................a small perturbation in the original signal prior to the EMD calculations. However, due to thedifference in the frequency ranges in which the various IMFs lie, small time-varying perturbationsignals of frequencies in these distinct ranges are needed. For each frequency range, the amplitudeof the perturbation needs to be orders of magnitude smaller than that of the corresponding IMF.For example, if for a normal EEG data segment (denoted by y[i]), there are six IMFs and theirfrequencies are about 5 kHz, 2 kHz, 1 kHz, 500 Hz, 200 Hz and 100 Hz, respectively (for IMFs 1–6), wewill need to add the following small sinusoidal signals: y[i] = y[i] + 0.9 × sin (2π i/(12 207/100)) + 0.5 ×sin (2π i/(12 207/200)) + 0.25 × sin (2π i/(12 207/500)) + 0.125 × sin (2π i/(12 207/1000)) + 0.0625 × sin(2π i/(12 207/2000)) + 0.03 × sin(2π i/(12 207/5000)), where the amplitudes of the IMFs are typically larger than10. The perturbation signals thus will not have any practical influences on the IMF results for normaldata. However, when there is a discontinuity with a linear relaxation in time, the corresponding IMFswill contain the added small sinusoidal oscillations instead of generating divergence or large anomalies(figure 11g–l). In addition, when the original data is contaminated by a small segment of zeros, withoutadding the small oscillations, the resulting IMFs will oscillate wildly in this region with amplitudesorders of magnitude larger than those of the normal datasets (figure 11a–f ). This is because, for obtainingeach IMF, EMD looks for the local maxima and local minima and then approximates the data with cubicspline connecting the maxima or minima. When a segment of zeros is encountered, there are no localmaxima or minima so that the EMD extrapolates with cubic spline using the maxima or minima outsidethis region. For the first IMF, as the frequency is the highest (approx. 5 kHz), even a zero segment ofabout 0.1 s would correspond to about 500 maxima or minima. Thus, the extrapolation will generateextremely large, artificial oscillations. The remainder obtained by subtracting IMF 1 from the originaldata will compensate the large oscillations in IMF 1, but they will propagate to subsequent IMFs. Theconclusion is that, adding the small sinusoidal perturbing signals causes essentially no difference in theoriginal signal (about 1 part in 1000), but the artificial anomalies can be effectively eliminated.
Ethics. This study was conducted in accordance with Federal and University of Florida Institutional Animal Care andUse Committee policies regarding the ethical use of animals in research (IACUC protocol D710).Data accessibility. All data used in the study have been uploaded onto Google Drive and are publicly available:https://drive.google.com/drive/folders/0B7S5nQOU−nMIYkEyYmJvTGpNOVU?usp=sharing.Authors’ contributions. Y.-C.L., L.H., P.R.C., W.L.D. and M.S. conceived and designed the research. The data were acquiredin P.R.C.’s laboratory. L.H. and X.N. developed the computational method and performed simulations. All analyseddata. Y.-C.L. and L.H. drafted the manuscript. All authors gave final approval for publication.Competing interests. We have no competing interests.Funding. The National Institutes of Biomedical Imaging and Bioengineering (NIBIB) through Collaborative Research inComputational Neuroscience (CRCNS) grant numbers R01 EB004752 and EB007082, the Wilder Center of Excellencefor Epilepsy Research and the Children’s Miracle Network supported this research. This work was also supported bythe US Army Research Office under grant no. W911NF-14-1-0504. L.H. was supported by NNSF of China under grantnos. 11135001, 11375074 and 11422541. Y.-C.L. acknowledges support from the Vannevar Bush Faculty Fellowshipprogram sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineeringand funded by the Office of Naval Research through grant no. N00014-16-1-2828.
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